252
'he envelope
;". I didn't
Ron Devi.
THE ONLY MAN INFINITY FEARS
not believe
A. ROSS ECKLER
Morristown, New Jersey
issue.
=s aplece:
:les in the
magazlne,
130
pages
$4.
biography
'1St. $8.
:i11 avail­
,een sold.
John Candelaria of Yucaipa, California is a man who leads two
lives. At work he is a mild-mannered Clark Kent, scientific illus­
trator and surveyor-at Caltrans
(occupied with traffic-flow studies
and highway planning); at home, he becomes the man of steel,
effortlessly vaulting past such mega-numbers as the googol (one
followed by one hundred zeros), the googolplex (ten raised to the
googol power), Archimedes' number (one followed by 80 quadrillion
zeros), and Graham's number (a number so immense that it takes
a page of specially-invented notation to define it). Since 1975,
Candelaria has been preoccupied - some might say obsessed - with
the task of creating a systematic nomenclature for large numbers.
In three earlier Word Ways articles (August 1975, February 1976,
May 1983) he described earlier stages of his work; this article
summarizes his completed edifice, which supplies mankind with all
the number-names that it could pOSSibly want.
Make no mistake about it, these number-names
are unimaginab­
ly large, far larger than is needed to describe, say, the number
of electrons that could be hypothetically employed to fill the uni­
verse. Even writing such numbers out is a task best left to the
imagination; in one bit of vivid imagery, Candelaria points out
that a googolplex could not be typed on sheets of paper forming
a cubic stack of 40 billion light-years on a side! Small wonder
that Isaac Asimov, upon reading the description of the Candelaria
system, moaned "your [Large Number Denomination] system makes
my head swim endlessly" and the Guinness Book of World Records
saluted him in its 1986 edition as "the only man infinity fears".
(Of course, Norris McWhirter is engaging in hyperbole; in reality,
Candelaria's largest number-name is
only an infinitesimal part
of aleph-null, to say nothing of the Cantorian structure of higher­
order infinities.
Lest Word Ways readers I heads swim as well, this article does
not attempt to describe Candelaria's achievement in full detail.
It is enough to appreciate the pillars on which it rests: first mul­
ti plica tion,
and next repeated exponentiation. Cardinal numbers
increase by addition: one plus one is two, plus one is three, etc.
Similarly, number-names increase by multiplication: one million
times one thousand is one billion, times one thousand is one tril­
lion, and so on. Candelaria calls this counting by periods, impli­
citly defined by (number of zeros) = 3(period) + 3. Unabridged
dictionaries define number-names up to a period of 20 (vigintil­
lion), and for good measure throw in a period of 100 (centillion).
253
Candelaria s first achievement \Vas to fill in the gap, creating
a nomenclature that, at least in principle, could describe every
period from one through one million: viginti-million with period
21, viginti-billion with period 22, etc. Just as the cardinal num­
bers become unwieldy to write out in words as one proceeds toward
one million, so do Candelaria's. The table below summarizes the
high spots, analogous to ten, one hundred, one thousand, etc.
I
Period
1 million (2 billion, 3 trillion, etc.)
10 decillion (20 vigintillion, 30 trigintillion, etc.)
100 centillion (200 ducentillion, 300 trecentillion, etc.)
1000 decingentillion (2000 bi-decingentillion, etc.)
10000 deci-decingentillion (20000 viginti-decingentillion, etc.)
100000 cent i-decingenti n ion
(200000 d ucen ti-decin"gentill ion, etc.)
1000000 mi11i11ion
At millillion, Candelaria shifted gears, using the concept of re­
peated exponentiation of the period to gain larger steps in his
nomenclature. The period of a number can be written in the expon­
ential form period = 103i +3 , where the index i is now the counter:
Index i
1 milli11ion (2 bi11illion, 3 trilli11ion, etc.)
10 decillillion (20 viginti11illion, 30 trigintillillion, etc. )
100 centi11illion (200 ducentilli11ion, etc.)
1000 decingentillillion (2000 bi-decingenti11i11ion, etc.)
10000 deci-decingentillillion, etc.)
100000 cen ti-dec ingen t illi 11 ion
1000000 mill ionei 11 ion
One can
now move to a second-level exponentiation in which the
period becomes period
10
3003i+3)+3
with a new index i as counter:
Index i
1 mill ionei 11 ion (2 bi 11 ionei 11 ion, 3 tri 11 ionei 11 ion, etc.)
deci11ioneillion (20 viginti11ionei11ion, etc.)
10
centillionei11ion
(200 ducenti11ioneillion, etc.)
100
decingenti11
ionei11
ion (2000 bi-decingentillionei11 ion, etc.)
1000
10000 deci-decingentill ionei 11 ion
100000 centi-decingentill ionei 11 ion
1000000 mi11itwoi11ion
It is now time to introduce a new counter, that of the number
of levels of exponents in the period. Millionei11ion can be written
in two ways: it is the last number-name with a period using a
single exponent level, and the first number-name with a period
using two exponent levels. Similarly, mi 11 itwoi11 ion is the di viding­
point between two and three exponent levels. One can at this point
3003i+3)+3
define a third exponent level, period = 10 300
) +3, carrying
the nomenclature from millitwoi11ion to mi11ithreei11ion, but Candel­
aria, realizin
of baby ones,
Level
1 mill
10 mill
100 mill
1000 milli
10000 milli
100000 milli
1000000 mill i'
Note that the
exponential mo
But now th
role that per
previously de
a mill ion tirr
it goes, like
feed on:
period 1:
level 1:
layer 1:
height 1:
elevation 1:
altitude 1:
perigee 1:
Now, if one
n umber-names
AN ANAG
Would y
anagran
GREAT (
ANGLO
Group (
gram fo
name ot
pie's ]v.
for oth,
Society,
for $5 I­
:ap, creating
~scribe every
with period
ardinal num­
)ceeds toward
mmarizes the
d, etc.
.)
etc. )
254
aria, realizing that it was time to start taking giant steps instead
of baby ones, let the exponent level provide the index:
Level
1 millioneillion (2 millitwoillion, 3 millithreeillion, etc.)
10 millitenillion (20 millitwentyilllon, etc.)
100 millihundredi11ion (200 millitwohundredillion, etc.)
1000 mi llithousandillion
10000 mill i tenthousandi 11 ion
100000 mi lli hundred thou sandi 11ion
1000000 mill im illi.on i 11 ion
In,
Note that the period of the final number-name is described by an
exponentia 1 mountain a million levels high!
mcept of re­
steps in his
n the expon­
But now the metamorphosis is complete; exponent level plays the
role that period did previously. One can, in analogy to what was
previously done with the periods, iteratively exponentiate levels
a million times, calling them layers to avoid confusion. And so
it goes, like DeMorgan' s famous
fleas that have greater fleas to
feed on:
etc.)
lllion, etc.)
le counter:
n, etc.)
:. )
n which the
i as counter:
, etc.)
ion, etc.)
the number
be written
od using a
h a period
Ie dividing­
t this point
l
+3
.
, carry~ng
but Candel­
period 1:
leve 1 1:
layer 1:
height 1:
elevation 1:
altitude 1:
perigee 1:
million
mi11ioneillion
mill illione i 11 ion
mill ionei 11 ionei11ion
millioneillim i 11ion i Ilion
mill ioneillim illi 11 im ill ion i Ilion
mill itwoi 11 im ill ioni 11 ion
Now, if one can only figure out
number-names ...
a way to index this sequence of
AN ANAGRAM GENERATOR
Would you like a microcomputer program which finds all the
anagrams of a short phrase such as MARGARET THATCHER (a
GREAT CHARM THREAT) or RONALD WILSON REAGAN (the INSANE
ANGLO WARLORD)? The Boston Computer Society's Macintosh
Group (One Center Plaza, Boston MA 02108) offers ¥J.ih a pro­
gram for $10 plus $2 postage, called Ars Magna
. As this
name of the group suggests, it runs at present only on Ap­
ple 1 s Macintosh computer, but may eventually be available
for other microcomputers. If one joins the Boston Computer
Society, one can buy this program, plus dozens of others,
for $5 plus $2 postage apiece.